Diagonalization

This article/section deals with mathematical concepts appropriate for a student in late high school or early university.

Diagonalization is a technique first used by Georg Cantor, a Germanmathematician. He used it to show that the real numbers can not be put into one-to-one correspondence with the natural numbers, thereby demonstrating the real numbers are not countable. This method can also be applied in other contexts, to show that two sets can't have a correspondence. For example, it can be used to show that no set can be in 1-1 correspondence with the set of all of its subsets.

Proof of the non-countability of real numbers

First, we create a 1-1 correspondence between the entire real line and the open interval . This function:

maps the entire real line to the open interval . Its inverse:

maps the open interval to the entire real line.

This means that the real numbers are in 1-1 correspondence with the natural numbers if and only if the open interval is in correspondence.

Assume the numbers in this open interval are in a 1-1 correspondence with the natural numbers. Then we can make an (infinite) sequential list of them, like this:

Where

Construct the number,

, where

when and when .

Therefore is not in the list, so we have a contradiction and our assumption is false, the numbers in are not countable. Therefore is uncountable.[1]

Diagonalization and the Existence of God

Some have cited diagonalization as a formal challenge to Saint Anselm's ontological argument for the existence of God. In summary, Anselm argued that there must be a greatest idea and what could be greater than God? Therefore God exists.[2]

However, diagonalization argues that no greatest idea can exist: quite bluntly, God is infinite, therefore He can be diagonalized to produce an even greater infinite.[3]